If `(1)/(sqrt2) lt x lt 1`, then prove that `cos^(-1) x + cos^(-1) ((x + sqrt(1 - x^(2)))/(sqrt2)) = (pi)/(4)`

Explanation

`cos^(-1) x + cos^(-1) ((x + sqrt(1 - x^(2)))/(sqrt2))`
Let `cos^(-1) x = theta " or " x = cos theta`
For `(1)/(sqrt2) lt x lt 1, 0 lt theta lt (pi)/(4)`
`rArr cos^(-1) x + cos^(-1) (x + sqrt(1 - x^(2))/(sqrt2))`
`= theta + cos^(-1) ((cos theta + sqrt(1 - cos^(2) theta))/(sqrt2))`
`= theta + cos^(-1) ((cos theta + sin theta)/(sqrt2))`
`= theta + cos^(-1) (cos (theta - (pi)/(4)))`
Now, `(theta - (pi)/(4)) in (-(pi)/(4), 0)`, which is not the principal values of `cos^(-1)` function
But `((pi)/(4) - theta) in (0, (pi)/(4))`
`rArr theta + cos^(-1) (cos (theta - (pi)/(4))) = theta + cos^(-1) (cos ((pi)/(4) - theta))`
`= theta + ((pi)/(4) - theta)`
`= (pi)/(4)`